For a given expanding d-fold covering transformation of the one-dimensional torus, the notion of weak Gibbs measure is defined by a natural generalization of the classical Gibbs property. For these measures, we prove that the singularity spectrum and the $L^q$-spectrum form a Legendre transform pair. The main difficulty comes from the possible existence of first-order phase transition points, that is, points where the $L^q$-spectrum is not differentiable. We give examples of weak Gibbs measure with phase transition, including the so-called Erdös measure.